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Parabolic-hyperbolic splitting in support propagation for stochastic porous media equations

  • Max Sauerbrey
  • Joshua Utley
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Abstract

We develop a randomly-localized energy method to derive qualitative results on the support propagation of stochastic porous media equations with linear conservative noise. Unlike in previous works, where energies are localized by weighting them with a spatial bump function, we weight them with profiles which are solutions to stochastic transport equations. This modulates out the support propagation due to the conservative noise term, and energy arguments---which otherwise fail in this setting---are again applicable. As a result, finite speed of propagation along with sufficient and necessary conditions on the existence of waiting time phenomena (modulo stochastic transport) are proven. These methods and results demonstrate, on short time scales, that the support propagation may be disintegrated into two independent parts, one due to the evolution of the porous media equation and one due to stochastic transport.

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Cite

BibTeXRIS
BibTeX
@misc{sauerbrey2026parabolichyperbolic,
  title = {{Parabolic-hyperbolic splitting in support propagation for stochastic porous media equations}},
  author = {Max Sauerbrey and Joshua Utley},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11468},
  eprint = {2609.11468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11468},
}
RIS
TY  - UNPB
TI  - Parabolic-hyperbolic splitting in support propagation for stochastic porous media equations
AU  - Max Sauerbrey
AU  - Joshua Utley
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11468
UR  - https://arxiv.org/abs/2609.11468
AB  - We develop a randomly-localized energy method to derive qualitative results on the support propagation of stochastic porous media equations with linear conservative noise. Unlike in previous works, where energies are localized by weighting them with a spatial bump function, we weight them with profiles which are solutions to stochastic transport equations. This modulates out the support propagation due to the conservative noise term, and energy arguments---which otherwise fail in this setting---are again applicable. As a result, finite speed of propagation along with sufficient and necessary conditions on the existence of waiting time phenomena (modulo stochastic transport) are proven. These methods and results demonstrate, on short time scales, that the support propagation may be disintegrated into two independent parts, one due to the evolution of the porous media equation and one due to stochastic transport.
ER  -